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The trajectory tracking for a class of uncertain nonlinear systems in which the number of possible states is equal to the number of inputs and each input is preceded by an unknown symmetric deadzone is considered. The unknown dynamics is identified by means of a continuous time recurrent neural network in which the control singularity is conveniently avoided by guaranteeing the invertibility of the coupling matrix. Given this neural network-based mathematical model of the uncertain system, a singularity-free feedback linearization control law is developed in order to compel the system state to follow a reference trajectory. By means of Lyapunov-like analysis, the exponential convergence of the tracking error to a bounded zone can be proven. Likewise, the boundedness of all closed-loop signals can be guaranteed.

During the last two decades, the control of systems using artificial neural networks (ANNs) has emerged as an effective and successful alternative to the conventional control techniques. The success of this approach lies on the universal approximation capability of ANNs which avoids the need for very time-consuming first principles modeling. Thus, it is possible to handle a broad class of nonlinear uncertain systems with little or (ideally) no a priori information.

The first deep insight about the identification and control of dynamic systems based on neural networks was provided by Narendra and Parthasarathy in [

Note that for the case of systems with multiple inputs, in particular when the number of states is equal to the number of inputs, the avoidance of the singularity cannot be guaranteed only by maintaining the coupling matrix of the neural network (see (

The deadzone is a nonsmooth nonlinearity commonly found in many practical systems such as electrohydraulic systems [

A direct way of compensating the deleterious effect of the deadzone is by calculating its inverse. However, this is not an easy question because in many practical situations, both the parameters and the output of the deadzone are unknown. To overcome this problem, in a pioneer work [

All the aforementioned works about deadzone studied a very particular class of systems, that is, systems in strict Brunovsky canonical form with a unique input. In this paper, we consider a wider class of systems, that is, uncertain nonlinear systems with multiple inputs where each input is preceded by an unknown symmetric deadzone. This global system could be seen as formed by an unknown affine system (see (

Throughout this paper, we will use the following notation.

In this study, the system to be controlled consists of an unknown multi-input nonlinear plant in which each input is preceded by an unknown symmetric deadzone; that is,

Note that

The plant described by (

The

Although the

The model of the

Based on (

In this section, the identification problem of the unknown global dynamics described by (

Note that an alternative representation for (

Note that the structure for the sigmoidal function

Typically,

On a compact set

The input signal

By substituting (

It can be observed that by using the model (

Since, by construction,

Based on the learning laws (

If Assumptions

the identification error and the weights of the neural network (

when

where

when

where

First, let us determine the dynamics of the identification error. The first derivative of

Consider the following Lyapunov function candidate:

Note that the utilization of

In this section, an appropriate control law

The reference trajectory

By simultaneously adding and subtracting the terms

Now, let us define the tracking error

If Assumptions

the tracking error and the state of system (

the norm of the tracking error, that is,

where

In order to illustrate the strategy proposed in this paper, a simulation example is presented in this section. A second order nonlinear system (see (

Tracking process: reference trajectory

Tracking process: reference trajectory

Control signal

Control signal

In this paper, the exponential tracking for a class of nonlinear systems with unknown deadzones using recurrent neural networks was considered. Since physical model is not available, the neural networks are used to identify the unknown dynamics. The main novelty in this study is a systematic procedure for the modification of the learning laws of the synaptic weights in such a way that the avoidance of the control singularity can be guaranteed. This objective is achieved by continuously monitoring the determinant of the coupling matrix or more specifically the input weight matrix. By defining a threshold for the determinant of the input weight, a “dangerous” region next to the singularity can be established. When such region is reached, the learning process is immediately stopped. In this way, the invertibility of the coupling matrix is guaranteed. The effect of this modification on the identification error stability is rigorously studied by means of Lyapunov analysis. On the basis of the instantaneous mathematical model obtained by the identification process, a singularity-free feedback linearization control law is developed in order to compel the system state to follow a reference trajectory. By means of Lyapunov-like analysis, the exponential convergence of the tracking error to a bounded zone can be proven. Likewise, the boundedness of all closed-loop signals can be guaranteed. Certainly, the main attractiveness of the suggested approach is its simplicity. However, it must be mentioned that the turning off of the learning law could reduce the system performance. In fact, in such conditions, the control action becomes mainly proportional.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper. The first author would like to thank the financial support through a postdoctoral fellowship from The Mexican National Council for Science and Technology (CONACYT) under Grant 194775-IPN.